Vector bundles on plane cubic curves and the classical Yang–Baxter equation

Abstract

In this article, we develop a geometric method to construct solutions of the classical Yang-Baxter equation, attaching to the Weierstrass family of plane cubic curves and a pair of coprime positive integers, a family of classical r-matrices. It turns out that all elliptic r-matrices arise in this way from smooth cubic curves. For the cuspidal cubic curve, we prove that the obtained solutions are rational and compute them explicitly. We also describe them in terms of Stolin's classification and prove that they are degenerations of the corresponding elliptic solutions.

Torsion Free Sheaves on Weierstrass Cubic Curves and the Classical Yang–Baxter Equation

“…Lemma 3. 16. Let (X, A), (C, ω) be a geometric datum as at the beginning of this section and Z be the irreducible component of X containing C. Then the geometric genus of Z is at most one.…”

“…Polishchuk's approach was extended in the joint works of the first-named author with Kreußler [18] and Henrich [16]. In particular, the expression of triple Massey products in geometric terms was elaborated in detail and extended to a relative setting of genus one fibrations; more complicated degenerations of elliptic curves (like the cuspidal Weierstraß cubic) leading to rational solutions, were included into the picture.…”

Simple vector bundles on a nodal Weierstrass cubic and quasi-trigonometric solutions of the classical Yang–Baxter equation

“…We begin with a quick review of the geometric theory of the classical Yang-Baxter equation (1), following the exposition of [11]; see also [15,24,25,13,12]. For g 2 , g 3 ∈ C, let…”

“…On the other hand, solutions of (1) can be studied using methods of algebraic geometry; see [15,24,25,13,12,11]. Namely, let E = V u 2 − 4v 3 + g 2 v + g 3 ⊂ P 2 be a Weierstraß cubic curve for some parameters g 2 , g 3 ∈ C. Such a curve is singular (nodal or cuspidal) if and only if g 3 2 = 27g 2 3 ; in this case E has a unique singular point s. Assume that A is a locally free coherent sheaf of Lie algebras on E such that: (13) H 0 (E, A) = 0 = H 1 (E, A) and A x ∼ = g for any smooth point x ∈ E.…”

“…Summing up, the geometric r-matrix attached to the pair (E, A), defines a non-degenerate skew-symmetric solution r (E,(n,d)) of (1) for the Lie algebra g = sl n (C), whose type is fully determined by the type of the underlying Weierstraß curve E and a natural number 0 < d < n, which is mutually prime to n. It is therefore a very natural problem to determine the corresponding solutions of (1) explicitly. It turns out, that for an elliptic curve E, one gets precisely the elliptic solutions of Belavin [3], where a choice of 0 < d < n mutually prime to n corresponds precisely to a choice of a primitive n-th root of 1; see [12,Theorem 5.5]. For a cuspidal curve E, one gets a distinguished rational solution of (1), whose combinatorics (in the sense of the works of the third-named author [28,29]) was determined in [12,Theorem 9.8].…”